e^x(1+x)/(cos(xe^x))^2

Question

find the integration

Answer 1

Question: Find the integral of e^x(1+x)/(cos(xe^x))^2

We can see the answer to this by observation.

Write the integrand as v.sec^2(u)

And we note that v {x(1+e^x)} is the diffferential coefficient of u {x.e^x}. i.e. v = u' = du/dx.

And since sec^(u) is the differential coefficient of tan(u), then

d/dx(tan(u)) = u'.sec^2(u) and u' = du/dx = v = x(1+e^x)

So, int e^x(1+x)/(cos(xe^x))^2 = tan(x.e^x)